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CERTAIN  PARTIAL  DIFFERENTIAL  EQUATIONS 


CONNECTED 


WITH  THE  THEORY  OF  SURFACES 


DISSERTATION 


SUBMITTED  TO  THE  BOARD  OF  UNIVERSITY  STUDIES  OF  THE  JOHNS  HOPKINS  UNIVERSITY 
FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


BY 

NATHAN  ALLEN  PATTILLO 


1897 


Of  THI 

VNIYER8ITY 


C^e  Motb  (0afttmore  (press 

THE  FRIEDENWALD  COMPANY 
BALTIMORE,  MD.,  U.  S.  A. 


^5 


«x 


INTRODUCTION. 

In  two  notes  in  the  Coraptes  Rendus  of  Oct.  26,  1896,  and  Nov.  16,  1896, 
and  in  the  American  Journal  of  Mathematics  of  Jan.,  1897,  Professor  Craig  has 
considered  certain  partial  differential  equations  connected  with  the  theory  of 
surfaces. 

In  this  paper  the  work  done  there  has  been  continued.  The  first  section  is 
introductory,  and  gives  a  brief  resume  of  Laplace's  method  for  the  integration  of 
linear  partial  differential  equations  of  the  second  order. 

In  the  next  two  sections  Professor  Craig's  equations  {Eoi)  and  (£"02) 
are  deduced  together  with  their  equivalent  equations  and  the  corresponding 
Laplace's  series,  and  the  general  formulae  for  calculating  their  invariants  are 
derived.  The  form  of  substitution  required  to  pass  from  an  equation  of  one 
series  to  one  of  the  other  series  has  also  been  found. 

In  the  fourth  section  it  is  shown  that  by  making  the  two  equations  {Eoi) 
and  {Eiiz)  identical  the  lines  of  curvature  form  an  isothermal  system  and  the 
reciprocals  of  the  two  principal  radii  of  curvature  satisfy  adjoint  equations. 

The  next  two  sections  are  taken  up  with  the  calculation  of  the  coefficients 
of  the  general  transformed  equations  and  making  the  general  equations  equal. 
Here  the  linear  element  reduces  to  the  isothermal  form. 

In  the  seventh  section  the  hypothesis  6oi  =  ao2  =  0,  has  been  made.  Here 
the  equations  (^01)  and  {E02)  reduce  to  integrable  forms  and  the  lines  of 
curvature  form  an  isothermal  system. 

The  following  section  has  been  devoted  to  the  consideration  of  the 
hypothesis : 

«oi  =  ^02 ) 

<^02  ^^  f^Ol' 

Then  the  conditions  have  been  found  that  must  exist  in  order  that  the 
reciprocals  of  the  principal  radii  of  curvature  of  parallel  surfaces  may  satisfy  the 
same  partial  differential  equations  as  those  of  the  original  surface. 

The  forms  to  which  the  equations  {Eoi)  and  (Eoz)  reduce  for  the  ellipsoid 
and  the  cyclide  of  Dupin  are  then  given  and  it  is  seen  that  they  may  be  readily 
integrated  by  the  method  considered  by  Poisson. 

I  desire  to  make  this  acknowledgment  of  gratitude  to  Professor  Craig,  who 
suggested  to  me  the  subject  of  this  dissertation  and  to  whom  I  am  indebted  for 
valuable  assistance. 


4  A  iCi^a 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/certainpartialdiOOpattrich 


CERTAIN  PARTIAL  DIFFERENTIAL  EQUATIONS  CONNECTED  WITH 
THE  THEORY  OF  SURFACES. 


1.     Consider  the  linear  partial  differential  equation 

|!f_  +  a|i'  +  6|f  +  0i.  =  0, 


(1) 


where  a,  b,  c  are  any  functions  whatever  of  u  and  v ;  it  can  be  put  in  either  of 
the  two  following  forms  : 


du 


Introduce  the  two  functions  h  and  k  defined  as  follows : 


(2) 


(3) 


A  =  ^ — \-  ab  —  C) 
du 

k  =  ~4-ab  —  c . 
dv 

If  h  is  zero,  the  first  of  equations  (2)  can  be  written 

Therefore,  the  complete  determination  of  all  the  integrals  of  equation  (1)  is 
reduced  to  the  successive  integration  of  two  equations  of  the  first  order 


(4) 


integration  which  requires  only  quadratures. 

In  like  manner,  if  k  is  zero,  the  second  of  equations  (2)  can  be  put  in 
the  form 


or  TH£  X 

UNiVERSITY   i 


6  Certain  Partial  Differential  Equations 

and  we  can  replace  equation  (1)  by  the  two  equations 

1^      +bf      =<p_, 


(5) 


Suppose  now  that  h  and  k  are  not  zero.  We  shall  show  that  these  functions 
enjoy  properties  of  invariance,  which  have  a  great  importance  in  the  study  of 
the  proposed  equation.     Consider  successively  the  substitutions 


<p  =  If' ; 

U=zf{u'),       VZ=(/f{l/)', 


Make  at  first 


X  being  any  function  of  u  and  v.     The  equation  in  <p  will  become 


(6) 


(7) 


If  we  calculate  the  new  values  of  h  and  ky  we  find  that  these  functions  have 
not  changed. 

Make  now  the  substitution 


u 


=f{u'),     v  =  ip{y^. 


Equation  (1)  will  become 

^li- + «i*'  w  aS + v  {»')  1^ + «/  M  (^  (»-)  f = 0 

The  new  values  A',  ^  of  A  and  fc  will  be 


Finally  the  substitution 


h'  =  hf{u').p'{v/),i 
k'  =  kf{u')iP'{i/).S 


(8) 


Connected  with  the  Theory  of  Surfaces.         7 

will  exchange  the  values  of  h  and  h.     All  these  properties  justify  the  name 
of  invariants  which  we  shall  give  to  h  and  h. 

Suppose  that  the  invariants  h  and  h  are  not  zero.     Make  the  substitution 
given  by  the  equation 

^i  =  |f  +  «^.  (9) 


Equation  (1)  can  now  be  written 


|g  +  %  =  %.  (10) 


If  we  eliminate  f  between  these  two  equations  we  get 


where 


|¥l  +  a,|^^  +  6,|^-^  +  c,^,  =  0,  (11) 

dlosh     j^       7  „      3a  ,  86      ,3  log  A,  ,^^ 

dv  du     dv  dv 


The  value  of  the  invariants  will  be 

K=2h  —  k        ^J|^   ,1  ^^3^ 

k,  =  h.  J 

They  are  expressed  only  as  functions  of  the  invariants  of  the  given  equation. 
Make  also  the  substitution 


We  shall  have 


^_,  =  ||  +  6^.  (14) 

^  +  «f'-i  =  %,  (15) 


and  we  get  the  following  equation  in  ^_i 

&^  +  °-.%-+*-.%-  +  <'-*-.  =  0-  (16) 


In  this  the  coefficients  have  the  values 


«_,  =  «,    6_,  =  6-^^^    c,=:c-i^,  +  ^-a^\         (17) 


du    '  dv      du  du 


8  Cebtain  Partial  Differential  Equations 

The  values  of  the  invariants  will  be 


k^i=2k  —  h 


logk^  > 
hdv  '  J 


(18) 


Thus  we  deduce  from  the  proposed  equation  (E)  two  new  equations  (Ei)  and 
{E_i).  We  can  apply  the  same  method  to  these  two  equations,  but  we  will 
not  get  two  new  equations  for  each  of  them.  The  first  of  the  two  substitutions 
applied  to  the  equation  (^_i)  will  give  us   the   proposed  equation  in  which 

^  will  be  replaced  by  -^ .     The  second  substitution,  applied  to  (Ei),  will  bring 

us  back  to  the  proposed  equation  in  which  f  will  be  replaced  by  -^ . 

If  then  we  regard  as  equivalent  two  equations  which  reduce  to  each  othe 
by  changing  <f  into  X<p  and  which  have,  therefore,  the  same  invariants,  we  see 
that  the  substitutions  of  Laplace  applied  successively  will  give  only  one  linear 
series  of  equations 

....,(J5;_,),(^_i),(^),  (^0,(^2),.... 

with  positive  and  negative  indices. 

The  invariants  of  equations  {Ef)  are  deduced  from  each  other  by  the  repeated 
application  of  equations  (13)  and  (18).     We  find  thus 


Solved  with  respect  to  hi  and  ki,  these  formulas  give 


(19) 


ki :=:  2«<4. 1  —  hi^^ 


aMogVhi 

dudv 


(20) 


We  can  give  to  i  all  integer  values,  positive  or  negative.  Instead  of  considering 
two  series  of  quantities  hi  and  ki  we  could  introduce  only  the  quantities  A< .  We 
shall  then  have  the  series 


.    .    .    .   >    ^_2>     "■— 1>     h  f     hi  f     hiy    .    .    .    ,  f 


Connected   with  the  Theory  op  Surfaces. 
deducing  one  from  the  other  by  the  formula 

A      4-  A      —2h—  3'MA< 


(21) 


The  invariants  of  the  equation  (Ei)  will  be  hi  and  hi_i.     By  the  linear  com- 
binations of  equations  (21)  we  can  obtain  the  relation 


hi^,  =  hi+h-k-^'^''^l\"  •  '^^ . 
^  dudv 


(22) 


2.  Let  u  and  v  denote  the  parameters  of  the  lines  of  curvature  of  a  surface, 
Pi  and  />2  the  principal  radii  of  curvature  of  the  surface  at  the  point  (w ,  r) ,  pi 
being  the  principal  radius  corresponding  to  the  line  v=z  const,  (the  w-line)  and 
P2  corresponding  to  m=  const,  (the  «-line) .  Let  i?i,  R^  denote  the  radii  of  geo- 
desic curvature  of  the  lines  u  =.  const,  and  v  =  const,  respectively. 
We  have  now 

_E  _G 


1^ 

R^ 

1 


—  1     dE 


2E^G  dv  ' 
—  1    dG 


El  ~  2Ga^E  du 


(23) 


We  have  also   the  following   equations   connecting   all   of  the   preceding 
quantities 


p{  dv       R2   \pi     Pz/ 

^2_VEf^_l_\_Q 
du       R^  \p^      p^J 


1  dp 

P2 


(24) 


Liouville's  formula  for  the  measure  of  curvature  can  be  written 


^/EG_  d  sfGt       d  s/E 


piPi       du  Ri    '  dv  R2 


(25) 


Substituting  the  values  of  pi  and  p^  in  equations  (23),  they  can  be  reduced 
to  the  form 

d  \/E  a/Ea/G' 


dv  pi^ 
d  s/G 


R2        p2 

VG\/E 


du  p2  Ri    pi 


(26) 


10 


Certain  Partial  Differential  Equations 


Differentiating  the  first  of  equations  (24)  for  u  and  the  second  for  v  and 

eliminating  —  and  —  respectively,  we  obtain  the  equations 
Pi         Pi 


d'    I       VG  d   I 


dudvpi       B-i  dupi      \du  ^^  R2  "^  Jii  J  dv  pi 
dudv  (h      \dv    ^  El    *    R2  Jdu  pz       Ri  dvp2 

That  is,  -  and  -—  are  particular  integrals  of  the  differential  equations 
Pi         Pi 

/  7^  >  _  3^f  01       ^G  8f  01       /  a  1      ^/G   .   VE\  9f  01  _  0 


(27) 


(28) 


The  differential  equation  satisfied  by  the  coordinates  {x,  y,  z)  of  the  point 
(u,  v)  of  the  surface  may  now  be  written 


d^<Po    ■  VG  d<po      ^/Edfii  _  Q 
dudv*    B2  du    '    Ri   dv 

From  equations  (26)  we  derive  in  the  same  way  as  above  the  following 


dudv      du  ^R,    dv        R,R.,^''~    ' 

.2^o2__a  1   y'^afo2_v^^  --0. 

dudv      dv  ^  R,    du        R,R^  ^"^ 


(29) 


(30) 


These  equations  have  ^-—   and  — —  respectively  as  particular  integrals. 

pi  Pz 

They  are  equivalent  to  equations  (28) ;  that  is,  they  have  the  same  invariants. 

3.   We  shall  now  consider  Laplace's    series   of  equations   corresponding  to 
(£"01)  and  {E^)y 

. .  . . ,  (J57_ii) , .  .  .  .  (J5/_ii) ,  (£'01) ,  (J?ii) )  (-^21) ,  •  .  •  .  (-Eii) ,  ....  1        .gjx 
. . .  .,  (^_«),.  .  .  .(^_x2),  (^02),  (^12),  (^22),  .  .  .  .  [E^),  .  .  .  ./ 

Using  equations  (3),  we  find  the  invariants  of  (j^i)  are 


^/EG    .    __ 


dudv    ^    ^2     "^    R1R2  ' 


(32) 


Connected  with  the  Theory  of  Sukpaces. 
those  of  {Eo'i)  are 

The  application  of  formula  (22)  will  give  the  general  invariants, 
we  find  for  all  positive  values  of  i 

hn  =  h,_,  ,  +  ^^^  log  ^5 -  3^^  log  KA^ . .  .  .  ^-.  . 
^e  =  ^i-i,  2  —  ci  o   log    -^    KiKz'  '  •  -hi-uz' 

OUQV  Ki 

K\  — -  "•<— 1,1)  \ 

f^t2  =^  "i— 1,2*  f 

For  negative  values  of  i  we  have 

A_«  =A_,,_,,.,-^|^-log^^A_n/^_,,.  .  .  .A-a-i),i, 

ga  a/  6r  9^ 

A_«  =/l_„_,>.2  +  g,^3-^  log  -]R- -3^-3-  log /^02A-12.  .   .   ./^-(,-l).2 

=  ^-«-l).2-3^l0g-^^/l_12/l_22.   .   .   ./l_U-l,.2. 


11 

(33) 
Thus 


(34) 


(35) 


k. 


-a  —  "-  «+i),i)  \ 

-C  =  "'-(i+l),2'  J 

From  the  first  of  equations  (34)  it  may  be  seen  that 


(36) 


(37) 


Therefore, 


,    _s/EG         d'    1      \^G 

"•11  ^^^  ''■02  )  \ 

"'ll  =^  "01  ^^  n^02  •  J 

Consider  the  general  invariants.     Let  us  suppose 

""i  — 1,1  ^^  "■<  — 2,2  J 
then  it  may  readily  be  seen  that 


Hi "i-l,2»  \ 


(38) 

(39) 
(40) 


12 


Ceetain  Partial  Differential  Equations 


But  we  have  found  equation  (39)  to  be  true  for  i=z2; 
all  values  of  i.  Therefore,  we  have  the  same  invariants  in 
equations  (31),  each  invariant  in  the  first  series  being  equal 
invariant  in  the  second  series.  Then  the  equations  {En) 
equivalent.  Hence,  the  first  may  be  reduced  to  the  second 
function  (pn  by  ^fi-1,2  •  We  can  determine  ^  by  identifying 
the  two  equations.  By  repeated  application  of  equations 
general  coefficient  «= 

__  d  log h  Ai '  •  •  -^i-i.! 


then  it  holds  for 
the  two  series  of 
to  the  preceding 
and  {Ei_i^2)  are 
by  replacing  the 
the  coefficients  in 
(12)  we  find  the 


a  — 


dv 


—  n  31og/to2 A<_2.a_^  d  log/tn.   .   .   .^<-l.l 

Oj-1.2  — 002 %;  —aoz  — ^ , 


Oj_l,2  ^  O02  •  J 


(41) 


(42) 


The  formulas   of    identification   which   we    may   take  from   equation   (7) 
will  give 


j^  3  log  X 

Oi_i,  2  —  «.i  i ^^  f 


fe<-l,2=^*l  + 


d  log  X 


(43) 


Replacing  o,_i,2  and  a^  in  the  first  equation  by  their  values  taken  from 
(41),  we  get  the  equation 

d\osX_d.VG 

After  integration  it  may  be  placed  in  the  form 


log>l  =  log^  +  logCr. 


(44) 


By  substituting  the  values  of  6<_i,2  and  6,1  taken  from  (42)  the  second  of 
equations  (43)  may  be  reduced  to  the  form 


Hence, 


log  ;=  log  -^  4-  log  V. 


(45) 


Connected  with  the  Theory  op  Surfaces.  13 

By  comparing  (44)  and  (45)  it  may  be  seen  that  U  and  V  must  be 
constants.  Therefore,  neglecting  a  constant  factor,  which  would  disappear  after 
substitution,  we  may  take 


Ro 


\/G 
Hence,  if  in  (E^)  the  function  f^  be  replaced  by  — p-  ^,_i  2,weshall  find  (JS',_i,2)' 

\/E 
In  the  same  way  it  may  be  shown  that  replacing  ^_<2  in  {E_i2)  by  ^5- 

yj-^-i),!  will  give  the  equation  (^_(j_i),i) , 

4.     We  shall  now  consider  the  result  of  making  the  equations  {Eoi)  and  (£"02) 
identical.     By  examining  the  coefficients  it  is  evident  that  we  have  the  condition 


l-^^=^' 
|-^^='>- 


(46) 


The  integral  of  the  first  of  these  equations  may  be  written 

R2 
Substituting  the  value  of  '^-  from  the  first  of  (23),  we  have 

hence, 

E=U^V^.  (47) 

From  the  second  of  equations  (46)  we  get 

This,  combined  with  (23),  gives 

we  have  then 

G  =  UW^.  (48) 

The  linear  element  of  the  surface  may  now  be  written 


14  Certain  Partial  Differential  Equations 

or 

ds'  =  X  ( Udu'  +  Vdv') ,  (49 ) 

the   U's  being  functions  of  u  alone  and  the    V's  of  v  alone.     Then  the  lines 
of  curvature  form  an  isothermal  system. 

The  equations  (28)  both  have  now  the  form 

?<P        "^(^^^P^^^f  —  o  (K0\ 

which  admits  —  and  —  as  particular  integrals.     It  may  also  be  written 

Since  E-=.XXJ and  G=.IV ,  we  have 

dudv  dv      du  du      dv        ' 

The  equation  satisfied  by  [x,  y,  z)  of  any  point  (w,  v)  of  the  surface  is  now 

aVo  ,  3  Jog  \// 3^0  ,  3iog\/^  a^ 0  _  Q  ,g2^ 

If  we  have  a  linear  equation 

ai*at;+"au  +  ^^+'^=^' 

its  adjoint  will  be  of  the  form 

d'd  dd       ,dO,/         da      db\.      ^ 

In  equation  (51)  we  have 

_aa_a6_  3Mogv>i_a^iog;._Q 

*^     du      dv  dudv  dudv         ' 


hence, 


since  ^  =  f7j  Fj .     Then  it  follows  that  (52)  is  the  adjoint  of  (51). 


Connected  with  the  Theoey  op  Surfaces.        16 

Therefore,  if  u  and   v  are   the   parameters   of   lines   of  curvature   of  a 
surface  and  the  linear  element  can  be  reduced  to  the  isothermal  form 

ds^  =  k{Udu'  4-  Vdv-"),    {k=  U^V^) 

then  the  Cartesian  coordinates  {x,  y,  z)  satisfy  a  differential  equation  which 
is  the  adjoint  of  that  satisfied  by  the  reciprocals  of  the  principal  radii  of 
curvature. 

The  invariants  of  (51)  are 


Therefore, 
We  can  find 


dudv 


The  invariants  of  this  equation  are 

^  _  1  31ogA31og;_^  1 
^       4     du       dv  *[  .->. 

lc,  =  h.  J  (^^) 

Then  we  have 

hi  =  ki=zh=ik.  (56) 

Then  all  the  invariants  in  the  series  will  be  equal.     Therefore,  we  are 
led  to  the  consideration  of  only  one  equation,  (-£"01). 

5.    Make  now  the  equations  {En)  and  (E12)  identical.     From  equations  (12) 
we  obtain 

_   _      VG_dlogho^ 

"u  —  g —  ^ f 

Ml  dv 

dv    °  Bi         R^  dv 

Equating  these  coefficients,  we  have 

9  io„V(^_aiog/io2  ,-„. 

dv^''^~R;-~dv~~'  (^^) 

Making  6u  =  6i2  gives  us 


16  Certain  Partial  Differential  Equations 

We  have  also 


•-^-l^^+S-^l+l")'-^^'- 


Since 


log  ^  =  log  F', 


(59) 


"~aw  R,        dudv  ^  R 

^  -_?„wv;^  .  a\/G_a\/£' ,  v^aiog/io2 
^'—awai'  ^  Ri  "^aw  r^     dv  r,  '^  r,     bv    ' 

By  equating  these  values  and  taking  (58)  into  account  we  get 

aMog;ioi_  d  \^E 
dudv        dv  Ri 
But  from  (23)  we  see  that 

d  a/e  a*  ,      ,^ 

Hence, 

^log^/GK  =  0.  (60) 

The  integral  of  (58)  may  be  written 

Ri 
and  this  combined  with  (23)  gives 

_  aiogV-g_p 

dv 

Therefore,  by  integrating  this,  we  have 

E=  U^e-'"",  (61) 


the  integration  of  equation  (57)  will  give 

log  F+C7=  log  A02; 
consequently 

hoi=iU'V'.  (62) 

We  may  write 

A.,  =  ^=y'^.  (63) 


Connected  with  the  Theory  of  Surfaces.  17 

Equation  (60)  can  now  be  put  in  the  form 

^^\og^G-\-^log  V  +  ^  log  V^—  ^  logi2i  =  0. 
dudv  duov    *  dudv  oudv 

But  from  (58)  we  see  that  the  second  and  third  terms  of  this  equation  vanish.    It 
then  becomes 

log^=:0. 


It  is  evident  then  from  (33)  that 

"'M  —   r)  T>    y 
and  consequently 

/ifli  —  "'01  • 

Combining  this  equation  with  (63),  we  get 

h^  =  V'^=U'V';  (64) 

therefore, 

V^ TV 

From  (23)  we  see  then  that 

OU 
The  integration  of  this  equation  will  give 

G=  7x6-2^'.  (65) 

Since  hot  =  ^oi  =  U'V; 

we  obtain  by  differention 

d"  log  hoi__Q 
dudv 
But  it  has  been  found  that 

J^  log  ^^  =  0 ; 
and  by  substituting  this  in  the  preceding  equation  we  find 


18  Certain  Partial  Differential  Equations 

This  value  substituted  in  tiie  second  of  equations  (32)  will  reduce  it  to  the  form 

"1  —  "7?  D     • 

Clearly  we  have  then 

/Coi  —  /Cos  —  "01  ——  ""OJ  )  \^"/ 

that  is,  all  the  invariants  in  the  two  leading  equations  are  now  equal. 
The  linear  element  in  this  case  will  be  expressed  by  the  equation 

^2_g-2(£r+r)|-f/-^g.£r^^2^  Fxe'^du"].  (67) 

If  then  M  and  v  are  the  parameters  of  the  lines  of  curvature  of  a  surface  and 
if  the  two  first  derived  equations  in  the  two  series  of  Laplace  (31)  are  identical, 
that  is,  {En)=^(Ei2)',  all  the  invariants  in  the  two  leading  equations  become  equal 
to  each  other  and  the  lines  of  curvature  form  an  isothermal  system. 

6.  We  shall  now  consider  the  general  case  and  make  (En)  =  (Efy).  To  do 
this  it  is  evident  that  we  must  have  the  coefficients  respectively  identical  in  the 
two  equations. 

We  have  as  in  (41) 

Gil  ==.  an 5—  log  hoihn  .  .  .  .  A<_-.  i  , 

Qi,  =  003 ^  log  htJhi  .  .  .  .  A<_i,  2  , 

=  aoj  —  -j^  log  hiJiii  .  .  .  .  hfi. 

Equating  these  values  of  the  coefficients,  we  get  the  equation 

d  log  hn  _  ^         d  log  hf. 

But  „   __      ^/G 

"  —       ST  ' 

and  ^   _      /fa    ^^„VE.VG\ 

These  values  substituted  in  (68)  reduce  it  to  the  form 

^IogA..=  J,log^+|^Iog*„; 


(68) 


Connected  with  the  Theoey  of  Surfaces.  19 

therefore,  since  

we  obtain  the  equation 

This  is  the  first  equation  of  condition  for  the  identity  proposed. 
We  know  that 

6*1  =  ^00 
0<2  =^  ^02  ) 

then  making  6<i  =  6,2> 

gives  us  simply  6oi  =  ^02  • 

Substituting  the  values  of  these  coefficients,  the  equation  becomes 

which  is  our  second  equation  of  condition. 

By  linear  combinations  of  the  last  of  equations  (12)  we  may  obtain  the 
general  coefficient 

„  ^       8«oi  8«<-i,i    I  3^01 

c„_Coi— ^-....       -^^yT'^  dv 

+  ••••+  ~d^     ^'' -Wv ^  ^ 

Combinations  of  the  first  of  equations  (41)  will  give 

3«oi don dai-\  _  __    d^    jQg  ^\ 

du        du       '  '  '  '        du  dudv 

H-3^1ogASr^Air^...VM,      (72) 

since,  from  (23),  we  have 

^  =  --1-  logV^. 

It  may  readily  be  seen  also  that 
dv  ^  dv   ^ ^     dv     ~       dudv     ^  B\ 


+  al^^«g^^  =  -a4^^4-    (^3) 


20  Certain  Partial   Differential  Equations 

by  remembering  that 

If  these  values  in  (72)  and  (73)  be  substituted  now  in  equation  (71),  it  may  be 
put  in  the  form 

<^ 

^     log  S  +  5^l0g^-'^n^  •  •  •  •  hi-^.^ 


~^»^ d^ 


=  Cn 


dudv 


<— 1 
6'' 


'<'g^  +  3S5s"'g5p*"---'^' 


,    d  log  hoJiii ....  ^<_i,  1 

^'' di •. 


(74) 


But  it  may  be  seen  from  the  first  of  equations  (34)  that  proper  linear  com- 
binations will  give 


f— 3 


(75) 


If  we  substitute  this  value  in  equation  (74),  we  have  finally 

„       «  IZ.  7,  h       9    log  ^01^11  •     ■    ■     •  ^^ir-\,X 


(76) 


In  the  same  manner  it  may  be  shown  that  in  the  second  series  of  equations  (31) 
the  general  coefficient  is 


Cc 


^     I  ^        h  h   ^  logApgAia. . .  .  ^i-1,2 

Co2  -|-  «o2 "■<-l,2 O02 ^ 

—  r  -u/i       h  —h  3  log ^11^ ^ii 

Co2  -|-  /loi n,i  Oo2  ^-  f 


(77) 


by  remembering  the  relations  which  exist  between  the  invariants  of  the  two  series. 

Since 

if  we  place 


Ooi  —  6o2 , 
Coi=:C(a  =  0, 


Cft  =  Cc  f 


Connected  with  the  Theory  of  Surfaces.  21 

we  will  get  after  obvious  reductions  the  following  equation 

h  —h      —b  ?J2i^  — A  —h   —h  ^i^sAe  nsi\ 

'''01 "i-l,  1 Oqi    ^-- /loi "il  Ooi  pC  .  \lO) 

But  we  know 

z.       ^   _        a'   ,     V-E^ 

"'01 "01 ^5 — ^5~  log  —fY-  } 

and  from  (34) 

^<l ^i-l,  1  = g^  log  ^-  Aii/l2i ....  ^<_i,  1 . 

Putting  these  values  in  equation  (78),  it  may  now  be  written 

_.  siogAoi ,  ,  a  log /til  _^     ,70. 

On  account  of  the  second  equation  of  condition  (70)  we  have 

and  from  the  first  condition  for  the  proposed  identity  we  found 

a,log/^a  =  |,log^. 

Introducing  these  values  in  equation  (79)  above  and  making  slight  reductions, 
it  becomes 

Since 

the  last  equation  may  now  be  written  in  the  form 

^log  VG^Wn A<-i.i,  (80) 

which  results  from  making  the  coefficients  Cn  and  c^g  identical. 

The  integration  of  the  second  equation  of  condition  (70)  will  give 

log^  =  log7', 


22  Certain   Partial  Differential  Equations 


or 


=  v. 


B2 

But  from  (23)  we  have 

hence 

By  integrating  the  last  equation  we  get 

E=U,,-'\  (81) 

The  equation 

3^1ogA„  =  ^log-^ 
may  be  integrated  and  placed  in  the  form 

log  K,  ==  log  "^  +  log  W  =  log  V  +  log  W  , 

therefore 

h,,=  V'V'.  (82) 

Equation  (80)  may  be  written  as  follows  : 
5^  log  V<?  +  A- log  V J5^- 5^  log  i2, 

Bat   it   has    previously  been  shown  that  the  second  and  fourth  terms  of  this 
equation  are  equal  to  zero.     It  then  becomes 

From  (34)  we  know  that 

^ii  =  Vu  — a^log-^— 3^1og/tiAi....Ai-i.i. 
Hence,  by  combining  this  with  the  equation  above,  we  get 


Connected  with  the  Theory  of  Surfaces.  23 

But  we  have  found 

consequently 

h,^,,,=  U'V'.  (84) 

The  differentiation  of  the  last  equation  will  give 

gf3,,l«gA-M=0.  (85) 

and,  if  this  be  substituted  in  (83),  we  have 

Again 

Ai-M='^*-2,l-a^-^l0g-^-g^^l0gM21.  .  .  .^i-2.i; 

and,  consequently 

Comparing  this  with  equation  (85),  we  have 

5^1ogAi_2,i  =  0. 

(juov 


Substituting,  as  before,  (86)  now  beco  nes 


(87) 


Thus,  it  may  easily  be  seen  that  by  continuing  this  process,  annuling  one 
term  each  time,  we  are  left  finally  with  the  equation 

^^  log  ^c^r-  =  0 . 
dudv        -til 

Then  we  have 

"-11  ^  "-01  ^^^  "'02  • 

If  we  should  make  the  equations  (ii^i)  and  {Eyi)  identical,  it  would  give 

"01  =^  "'11  f 

Kqi  =  kii . 


24  Certain  Partial  Differential  Equations 

But  from  the  relations  of  the  Id  variants  in  the  two  series  we  have 

kii  =  kfjfi  f 
hence  A02  =  ^01  =  ^n  > 

»02  =  nJoi  ^^^  "-11  • 

Then,  by  making  the  equations  (^„)  and  [Ea)  identical,  we  find  that  all  the 
invariants  in  the  two  series  are  equal  and  we  are  led  only  to  the  consideration  of 


Since 
we  have 


^log'^=0. 
The  integrals  of  these  two  equations  may  be  written  in  the  form 

V^  —Tjv 

Substituting  the  values  of  ^^—  and  ^—  from  equations  (23),  we  get 

■til  -"2 

—       ^        ^^  —  TTV 

1       ^^=Cr,F2. 


The  elimination  of  —       ^=^  from  these  equations  will  give 

2v^6r 

or  V-,    dE_V^  dG 

Ui    dv'~   Vi    du' 


Connected  with  the  Theory  op  Surfaces.  25 

where  the  U'S  are  functions  of  w  alone  and  the  VS  ofv  alone. 
Write  now 

then  the  equation  above  becomes 

|(ra)  =  |(FG).  (89) 

But  this  is  a  necessary  condition  for  the  existence  of  an  exact  differential  equation 

{UE)du-{-{VG}dv  =  d(py  (90) 

where 

r  — 1  ^ 

Take  u  and  v  arbitrary  and  ^  will  have  to  determined. 
From  (23)  we  have 

and,  by  introducing  the  value  of  E  from  the  first  of  the  two  equations  above,  it 
becomes 


s/G       a  /    1,    ^, .  1.   d<p\ 


In  like  manner  we  can  find 


3m 


A^E 1  dudv . 

do 
The  products  of  these  two  equations  will  give 


^=iW  =  ^^'^>^--  m 


26  Certain  Partial  Differential  Equations 

If  we  place 

we  have  then 


W 


2  fuUi-'elu+  'ifvVi^dv 


Hence,  equation  (91)  may  now  be  placed  in  the  form 

\dudvj 8  log  IF  3  log  W 


d<pd<p 


du 


or 


"^    ^ du' du    ^ dv         du     '     dv 

7.  "We  propose  now  to  consider  the  result  of  making  6oi  =  «08  =  0, 
that  is 

du  ^  B,  ^  B,        ""^ 


d  1     \^jE     f^G      - 


Wv'^'^'b; -^ -B 


(92) 


(93) 


By   substituting  the  values  found  in  (23)  the  two  equations   above  easily 
reduce  to 

|logiJ,=0, 

|logiJ.=,0. 


The  integrals  of  these  equations  are 


We  may,  therefore,  write 


B,  =  f{v),\ 
B,  =  W{u).i 

1  _     1     dVG_.j 
B~A^EG    du   ~     ' 

1—     l_d^/E_.r 


(94) 


Connected  with  the  Theory  of  Surfaces.        27 

where    ?7  is  a  function  only  of  u  and  V  a  function  of  v  only.     By  eliminating 
we  obtain  the  equation 


s/£:g 


yd  s/G jjd  f^E 

du  dv    ' 


^{V^G)  =  -^{U^E).  (95) 

We  see  that  this  is  a  condition  for  the  exact  differential  equation 

(96) 
where  we  have 


We  find  as  before 


U^E)  du  -\-{V^G)dv  =  d(p, 

a/E=z 

1 

U 

d<p 
du' 

VG  = 

1 

r 

d(p 

-"2 

— 

dudv 
df   ' 
du 

^E 
B,    - 

— 

dudv 
df   ' 

ev 

itial  equation 

dudv 

d<p 
du 

d<p 
dv 

(97) 

The  integral  of  this  may  be  written 

^  =  -log(C^,+  FO.  (98) 

It  may  then  easily  be  seen  that 


E=  f^' 


2) 


The  linear  element  may  now  be  expressed  by  the  equation 


^^=iU,+  v,Y  [^  ^^'  +  -W  ^"']  ^^^^ 


28 


or 


Certain  Partial  Differential  Equations 

ds'z=:^{Udw'-\-Vdi^). 

A 


This  is  a  form  which  characterizes  isothermal  systems. 

Then,  if  u  and  v  are  the  parameters  of  lines  of  curvature  of  a  surface,  and  if 
the  radius  of  geodesic  curvature  of  the  line  w  =  const,  is  a  function  of  w  alone  and 
that  of  v=z  const,  of  v  alone,  the  lines  of  curvature  of  the  surface  will  form  an 
isothermal  system. 

The  leading  equations  of  the  two  series  now  become 


while  that  satisfied  hy  (x,  y,  z)  remains 

Since  ^X=Ui-j-Vi, 

we  obtain  a^G d  log^/E F{      3  log  a,JX 

A/E__d\ogVG_      U[      _aiogVA 

jSx  ~        au    ~ u^+v~    du    ' 

By  substituting  these  in  the  equations  above  we  get 

/  r  \  _  ^<foi      d  log  A^X  3y7oi  _  0  ~ 

(j^)^  ^9  I  a  log  v-^  a^i  aiog/y/i^  ?^=zo. 

^  dudv  dv       du  du       dv 

The  invariants  of  the  two  proposed  equations  are  now 


»0l  ^  "  > 


Ao2  =  0> 


log>v/>^ 


(100) 


(101) 


(102) 
(103) 


(104) 
(105) 


Connected  with  the  Theory  of  Surfaces.  29 

That  is,  they  have  their  invariants  equal  but  taken  in  opposite  order. 

8.  Next  we  shall  consider  the  result  of  making  the  coefficients  of  the  equa- 
tions (£"01)  and  (£"02)  equal  but  taken  in  opposite  order,  that  is 

Ooi  ^^^  "02  > 
(l02  — -  ^01  • 

By  substituting  their  values  we  have 


-^  ^""^  R^^  B^—  dv    ^  R,  ^  R,  J 


(106) 


(107) 


By  combining  these  we  get 

du  ^""^  R,  -  dv  '""^  B,  ' 

or  _^_9    VG^  — ^    a   VE 

sfQ  du     R.,  ~A/Edv    R^  ' 

Taking  into  account  the  first  of  equations  (106),  we  have 

8    a/G_   d    ^E 
du    R2        dv    Ri 

These  values  placed  in  equation  (107)  will  give 

>5^  log  A^E=l  -^  log  VC? . 

If  we  integrate  this  equation,  we  shall  get 

E=  G  .U.V,  (108) 

where  ?7is  a  function  of  w  only  and  F  of  v  only.     The  linear  element  can  now  be 
put  in  the  form 

ds'=G{UVdu''  +  dv'), 

or  ds'=GV{Udu^-\-V,dv'').  (109) 


30  Certain  Partial  Differential  Equations 

Then  the  lines  of  curvature  form  an  isothermal  system. 

9.  Let  us  now  consider  the  forms  to  which  the  equations  (£"01)  and  (£'02)  reduce 
for  surfaces  parallel  to  the  original  one. 

Since  the  surface  is  referred  to  its  lines  of  curvature  we  have  the  following 
values  for  parallel  surfaces: 


pt  =  p2  —  a,( 


(110) 


(111) 


where  a  is  a  constant  which  changes  as  we  pass  from  one  surface  to  the  next  one. 
We  have  now 


a    ^  =f  p'  \'d    1 

du    Pi       \pl-{-aJ  du    p\  ' 
do     pi        \pi-{-aJ  dv    p\ 


(112) 


By  differentiating  the  first  of  these  equations  for  v  and  the  second  for  u  we  get 

^   1  _r  pi  V  a' 


1^3 


\p\  +  a) 


_a_i  1 


L^(  pi\ 

dudv  pi       \pi-\-aJ  dudv  p^i  ~*    du  \pt-^aj  dv  />?. 


(113) 


By  substituting  these  values  from  (112)  and  (113)  the  equation 

A_I__^Ai rAiog^4-VAAl.  =  o 

dudv  pi        E2   du  pi      \du     ^  B2        Ml  J  dv   pi         ' 
can  be  put  in  either  of  the  two  following  forms : 

dwd»  p\       Life        dv     ^\t>':  +  a)jdu  f,', 

\du    ^  Ei   ^  Ri  J  dv  p\ 


^ 


'/>?       Ri    du  />?       Idu    ^~B2  ^  Bi 


_-#-log 

du     "= 


\pl  +  aj] 


dv  pl~ 


(114) 


Connected  with  the  Theory  of  Surfaces. 


31 


By  examining  these  we  see  that  in  order  that  these  equations  reduce  to  the 
same  form  as  the  original  equation,  we  must  have 


The  integration  of  these  will  give 


(115) 


(116) 


In  the  same  manner  as  above  the  equation  (£"02)  may  be  put  in  either  of  the  forms 


d^v  f^      L  dv  '""^  R;  ^  R,        dv  '""^  \{,l  +  a)  J  du  pi 


J 


dud 


^v  pi       V  dv    ^   R,  ^  R,  J  du    pi 


VE   d_  JL 
R^    dv    pi 


=  0, 


iR,        du'^'^Kpl  +  aJ]  dv    pi 
The  condition  that  these  reduce  to  the  same  form  as  (i/oa)  is,  as  before. 


(117) 


pl  =  (p{u), 


=  (p{u),  I 
=  (pi{v).S 


(118) 


Then  if  each  of  the  principal  radii  of  curvature  of  the  parallel  surface  is  a  func- 
tion of  only  one  of  the  parameters  u  and  v,  the  reciprocals  of  those  radii  will 
satisfy  the  same  partial  differential  equations,  (£"01)  and  (£"02) ,  as  the  reciprocals  of 
the  principal  radii  of  curvature  of  the  original  surface. 
From  (110)  we  have 


T'  ^ 


Then,  the  equations 


Pi  P 

\/Gr \/(to 


pi 


d  1  yjg  a  VE  s/EG    ^E^  Q  ^ 

dudv    pi       du  Rz    dv   pi  R1R2  '    Pi 

_?_  V<^      d^  s/GdVG  \^EG  yG_^ 

dudv   Pi       dv  Ri   du  p2  RiRi  '    p2         ' 


32  Certain  Partial  Differential  Equations 

do  not  change  form  when  we  pass  to  a  parallel  surface. 

10.  (a)  For  the  ellipsoid  we  have  ^='i£-^;::;^,      ^— ^(^— ^)^     Then 
equations  (^oi)  and  {E(^)  become 


^udv      2{u  —  v)  du       2{u  —  v)   dv  ' 

^02_ 3  9f02     I  1  9f02__Q  . 

dudv      2  (w  —  v)  3w       2  (w  —  v)   dv  ' 


(119) 


and  the  equation  satisfied  by  {x,  y,z)i8 


a^^o  ,      1     d(po_     1     d<po--Q^  Q20) 

dudv      '2{u  —  v)  du       2{u  —  v)  dv  '  ^      ^ 

We  can  obtain  a  great  number  of  particular  integrals  of  these  equations. 
For  example,  we  can  find  solutions  which  are  the  product  of  a  function  of  w  by  a 
function  of  v. 

If  a  denotes  a  constant,  we  write 

^01  =  (w  —  a)-^  {v  —  a)-i 

1 

\/{u  —  af  {v  —  a) ' 
<p^=:{u  —  a)-^{v  —  a)-i 
1 


Ai/{u  —  a){v  —  a)'^ 
By  using  a  known  property  of  these  equations  we  have  also 

^01  =  (^  —  u)~^(u  —  a)~^{v  —  a)* 

V  —  w  T  u  —  a 

^02  =  (v  —  u)-^  {u  —  a)i  {v  —  a)-* 
1        lu  —  a 

V  —  u  y  V  —  a* 

"We  can  find  new  solutions  by  operating  on  those  already  found  with  the  symbols 

-'i  +  '^li  +  i^  +  i"- 


Connected  with  the  Theoby  of  Surfaces.  33 

We  have  also  the  general  formulae 

'  f{x){x  —  u)-^  {v  —  x)~  '^  dx-\-{v  —  u)~  1  /  /i  {x){x  —  .w)~ '  (v  —  «)*  dx  , 

u  vu 

f{x)[x  —  w)~*  {v  —  x)~ i  dx  -\-{v  —  u)~  1  /  /i  {x){x  —  w)* [v  —  x)~ ^  dx , 

which  contain  two  distinct  arbitrary  functions.  In  the  same  way  we  can  obtain 
integrals  of  equation  (120).  The  integration  of  this  equation  will  determine  the 
surfaces  which  admit  for  spherical  representation  of  their  lines  of  curvature  a  sys- 
tem of  spherical  homofocal  ellipses.  If  u  and  v  denote  the  parameters  of  these 
ellipses  and  X,  [x,  v  the  coordinates  of  a  point  of  the  sphere,  we  have 

,2_(a--w)(a--v) 
'^  —  (a  —  6)(a  —  c) ' 

(6__w)(6--j^) 
^  — (6  — a)(6  — c)' 
2 _  (c--w)(c— j;) 
—  (c  —  a){G  —  h) ' 

and  the  values  of  ^,  fx,  v  will  satisfy  equation  (120).     The  solution 

^0  =  C-J-C'(M  +  V), 

will  give  the  surface  of  the  fourth  class  defined  by  the  equation 

a)?  -j-  b[j?  -|-  ei^ 

P—  2  ' 

where  ^,  //,  v  are  the  direction  cosines  of  the  normal,  and  the  coordinates  of  the 
point  of  contact  of  the  tangent  plane  have  the  values 

x  =  {p^a)X,     y=:{p—h)fx,     z=z{p  —  c)v. 
If  we  take  the  solution 


^0  =  G\/{u  —  a){v  —  a)  f 

we  obtain  the  surfaces  of  the  second  degree. 

Equation  (120)  also  admits  as  particular  solutions  the  five  pentaspherical 
coordinates  of  the  cyclides  defined  by  the  formula 


^^^^(a.-a)(«.-«)(a.-.)_        (^^  i,  2,  3,  4,  5). 


34  Certain  Partial  Differential  Equations,  etc. 

(b)  For  the  cyclide  of  Dupin  the  coefficients  of  the  linear  element  are 


G  = 


1 


^V{u  —  v)J' 
The  equations  {Eqi)  and  (jS'02)  now  become 


3Voi   I      1     dfo} 
dudv     u  —  V  du 

CPJPo 


=  0, 


1  3f02__Q 


dudv     u  —  V  dv 
while  that  satisfied  by  the  point  coordinates  is 


(121) 


(^0)  = 


3^0  _ 


dudv     u  —  V  du      u  —  V  dv 


=  0 


(122) 


We  can  obtain  any  number  of  algebraic  integrals  of  these  equations  ;^r 
example 

>oi  =  V  —  a , 


foi 


y?02  =  w  —  a, 
[u  —  vf 
'  (v  —  a)[u  —  af ' 


{u  —  vf 

'P^  —  {u  —  a){v  —  af' 


The  integration  of  (J^o)  gives 


^0 


U- 


In  fact,  by  making  the  substitution 


^0  = 


M V 


U V 


the  equation  {E^  is  transformed  into 


dudv 


LIFE. 

I  was  born  at  Loachapoka,  Ala.,  and  received  my  early  training  at  the  High 
School  at  that  place.  I  was  graduated  from  the  Southern  University  (Ala.)  with 
the  degree  of  Bachelor  of  Science  in  1887.  After  spending  one  year  as  principal 
of  a  high  school,  I  returned  to  my  Alma  Mater,  where  I  remained  two  years  as 
Instructor  in  Mathematics,  and  received  the  degree  of  A.  M.  in  1890.  Having 
spent  another  year  teaching,  I  entered  the  Johns  Hopkins  University  in  1891, 
where  I  pursued  graduate  courses  in  Mathematics,  Physics,  and  Astronomy. 
During  1892-'94  I  was  Professor  of  Mathematics  in  Millsaps  College  (Miss.).  In 
1894  I  returned  to  this  University.  I  have  attended  the  lectures  of  Professors 
Craig  and  Franklin  and  Drs.  Ames,  Poor,  Hulburt,  and  Chapman,  to  all  of  whom 
I  am  grateful  for  the  kindnesses  that  I  have  received  from  Jl 

AprU,  1897. 


MAY  2  1935 


MAR  21   1! 


APR  26    U 


^^^  ^St94o/f 


N0V2S 


''''^'^•3S 


/ 


"j'm 


■''•  ''?.#:■ 


^; 


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